Nonhomogeneous fractional p-Kirchhoff problems involving a critical nonlinearity

Nonhomogeneous fractional p-Kirchhoff problems involving a critical nonlinearity

Jiabin Zuo

, Tianqing An

, Guoju Ye

and Zhenhua Qiao

1College of Science, Hohai University, Nanjing 210098, P. R. China

2Faculty of Applied Sciences, Jilin Engineering Normal University, Changchun 130052, P. R. China

3School of Electronic and Information Engineering, Jiangxi Industry Polytechnic College, Nanchang 330099, P.R. China.

Received 15 March 2019, appeared 26 June 2019 Communicated by Patrizia Pucci

Abstract. This paper is concerned with the existence of solutions for a kind of nonho- mogeneous critical p-Kirchhoff type problem driven by an integro-differential operator L^p_K. In particular, we investigate the equation:

M Z Z

R²ⁿ

|v(x)−v(y)|^p

|x−y|^{n+ps dxdy}

L^p_Kv(x) =µg(x)|v|^q−2v+|v|^p∗s−2v+µf(x) inRⁿ,

where g(x) > 0, and f(x) may change sign, µ > 0 is a real parameter, 0 < s < 1 <

p < ∞, dimension n > ps, 1 < q < p < p^∗_s, p^∗_s = _n−ps^np is the critical exponent of the fractional Sobolev space W_K^s,p(Rⁿ). By exploiting Ekeland’s variational principle, we show the existence of non-trivial solutions. The main feature and difficulty of this paper is the fact thatMmay be zero and lack of compactness at critical levelL^p∗s(_Rⁿ)_. Our conclusions improve the related results on this topic.

Keywords: fractional p-Kirchhoff problems, non-homogeneous, critical nonlinearity, Ekeland’s variational principle.

2010 Mathematics Subject Classification: 35R11, 35J60, 35A15, 47G20.

1 Introduction and main results

In this paper, we study the existence of solutions of the following the fractional p-Kirchhoff type equation involving a critical nonlinearity:

M _{Z Z}

R²ⁿ

|v(x)−v(y)|^p

|x−y|^{n+ps dxdy}

L^p_Kv(x) =µg(x)|v|^q−2v+|v|^p∗s−2v+µf(x) inRⁿ, (1.1) where n > ps, p > 1 and s ∈ (0, 1), p^∗_s = _n−^np_ps, µ is a positive parameter, M : R⁺₀ → _R⁺₀ is a continuous function, where g(x) > 0, and f(x) may change sign onRⁿ, and L_K^p is the

BCorresponding author. Email: zuojiabin88@163.com

non-local p-fractional type operator defined as follows:

L_K^pφ(x) =2 lim

ε&0

Z

Rⁿ\Bε(x)

|φ(x)−φ(y)|^p−2(φ(x)−φ(y))K(x−y)dy for all x∈_Rⁿ, along anyφ∈C₀^∞(_Rⁿ)_{, where} B_ε(x)denotes the open ball inRⁿof radiusε>0 at the centre x ∈ _Rⁿ and K : Rⁿ\{0} → (0,∞) is a measurable function which satisfies the following properties:

(ζK∈ L¹(_Rⁿ), whereζ(x) =min{|x|^p, 1};

there existsκ₀ >0 such thatK(x)≥κ₀|x|^−(n+ps) for any x∈_Rⁿ\{0}. (1.2) WhenKis a standard type (i.e.K(x) =|x|^−(n+ps)). In this case, the problem (1.1) becomes

M Z Z

R²ⁿ

|v(x)−v(y)|^p

|x−y|^{n+ps dxdy}

(−4)^s_pv =µg(x)|v|^q−2v+|v|^p∗s−2v+µf(x) inRⁿ, (1.3) where(−4)^s_pis a fractional p-Laplace operator defined by

(−4)^s_pv(x) =2 lim

ε→0⁺

Z

Rⁿ\Bε(x)

|v(x)−v(y)|^p−2(v(x)−v(y))

|x−y|^{n+ps dy}

for x ∈ _Rⁿ. For more details about the fractional p-Laplacian, we refer to [6,7,12,21,25,26]

and the references therein. Moreover, if f(x) =0, then the problem (1.3) reduces to M

_{Z Z}

R²ⁿ

|v(x)−v(y)|^p

|x−y|^{n+ps dxdy}

(−4)^s_pv=µg(x,v) +|v|^2∗s−2v inRⁿ. (1.4) In [27], Zhanget al. obtained infinitely many solutions for the problem (1.4) by using Kajikiya’s new version of the symmetric mountain pass lemma.

In recent years, fractional and nonlocal problems have received extensive attention, espe- cially involving critical nonlinear terms. For instance, in bounded domains, we refer to [11,18];

in the whole space, see [13]. It is worth pointing out that the interest in nonlocal fractional problems is beyond the curiosity of mathematics. Indeed, there is much literature on nonlocal operators and their applications, here we list only a few, see for example [14,16,19,28,29] and the references therein. For the basic nature of Sobolev spaces, we recommend readers to read the literature [15,17].

Recently, in [10], Fiscella et al. proposed a stationary Kirchhoff variational equation and investigated a model given by the following formulation:





 M

_{Z Z}

R²ⁿ

|v(x)−v(y)|²

|x−y|^{n+2s dxdy}

(−4)^sv=µg(x,v) +|v|^2∗s−2v inΩ,

u=0 inRⁿ\_Ω,

(1.5)

whereM = a+bt for allt ∈_R0⁺_{, here}a > _0,b≥ 0. Kirchhoff problems like (1.5) are said to be non-degenerate ifa>0 andb≥0, while it is named degenerate ifa=0 andb>0. For the two separate cases, we also refer to [3,20] about non-degenerate Kirchhoff type problems and to [9,22] about degenerate Kirchhoff type problems for the recent advances in this direction.

For non-homogeneous cases, in [23] Xiang et al. showed the multiplicity of solutions for the nonhomogeneous fractional p-Kirchhoff equations involving concave-convex nonlineari- ties by using the mountain pass theorem and Ekeland’s variational principle. Taking the same

approach as in [23], Chenet al.[4] studied the multiplicity of solutions for a kind of Dirichlet boundary value problem in bounded domain. Furthermore, Pucci et al. in [21] investigated the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type in Rⁿ.

Inspired by the above papers, we will study the existence of solutions for nonhomogeneous fractional p -Kirchhoff problems involving critical nonlinearities and weight terms. As far as we know, there are no works on the the problem (1.1). Thus, our conclusions are new on this topic. To this purpose, we first suppose that

(M₁) There existsθ ∈ 1,_n−ⁿ_ps

, such that θM(f t) =θ

Z _t

0

M(s)ds≥ M(t)t, ∀ t∈_R⁺₀;

(M₂) for everyσ>0 there existsm=m(σ)>0 such thatM(t)≥ mfor all t≥σ;

(M₃) there existsa₀ such thatM(t)≥a₀t^θ−1 for allt∈ [0, 1].

At present, the assumptions about Kirchhoff functions Mare diverse; we refer the interested reader to [5,10] and references therein. Here we do not require Kirchhoff functions to satisfy any monotonicity, and our paper covers the degenerate case.

Next, concerning the positive weightg:Rⁿ →_R, we assume that (Z₁) g∈ L^q1(_Rⁿ)∩L^∞_loc(_Rⁿ), f ∈ L^ξ(_Rⁿ), withq₁= _p∗^p∗s

s−q,ξ = _p∗^p∗s s−1. (Z₂) 1<q< p <θp< p^∗_s.

Before we present the main results, we give some notations. The function spaceW_K^s,p(_Rⁿ) denotes the closure of C^∞₀ (_Rⁿ), andW_K^s,p(_Rⁿ)is a Banach space which can be endowed with the norm, defined as

kφk_s,p = Z Z

R²ⁿ|φ(x)−φ(y)|^pK(x−y)dxdy ¹_p

, for all φ∈C₀^∞(_Rⁿ)_.

Now we define weak solutions for the problem (1.1):

Definition 1.1. We say thatu ∈W_K^s,p(_Rⁿ)is a weak solution to the problem (1.1), if M(kv(x)k_s,p^p )

Z Z

R²ⁿ|v(x)−v(y)|^p−2(v(x)−v(y))(φ(x)−φ(y)K(x−y)dxdy

=µ Z

Rⁿg(x)|v(x)|^q−2v(x)φ(x)dx+

Z

Rⁿ|v(x)|^p∗s−2v(x)φ(x)dx+µ Z

Rⁿ f(x)φ(x)dx, for any φ∈ X₀.

The main results of this paper are as follows.

Theorem 1.2. Set K:Rⁿ\ {0} →(0,∞)be a function fulfilling(1.2) and if(M₁)–(M₃)and(Z₁)– (Z₂)hold. Then, there existµ₀,τ₀>0such that for anyµ∈(0,µ₀), the problem(1.1)has at least one non-trivial solution with negative energy in W_K^s,p(_Rⁿ)whenkfk_ξ ≤τ₀.

Theorem 1.3. Assume that all conditions in Theorem 1.2 are fulfilled. Then, there existµ⁰₀, τ₀⁰ > 0 such that for anyµ∈(0,µ⁰₀), the problem(1.1)has at least one non-trivial non-negative solution with negative energy in W_K^s,p(_Rⁿ), provided that f ≥0a.e. inRⁿandkfk_ξ ≤ τ

0 0.

Remark 1.4. The main novelty of our paper is that we discuss the problem (1.1) containing a critical nonlinearity, which is not considered in previous references, such as [23]. To overcome this difficulty about lack of compactness at critical levelL^p∗s(_Rⁿ), we fix parameterµunder a suitable threshold strongly depending on conditions(M₂)and(M₃).

Finally, we give a simple example to show a direct application of our main results.

Example 1.5. Letn>1 andθ>1. We consider the following the problem a+b

_{Z Z}

R²ⁿ

|v(x)−v(y)|²

|x−y|^{n+1 dxdy} θ−1!

(−4)¹²v(x) =µg(x)v^q−1+vⁿⁿ⁺⁻¹¹ +µf(x) inRⁿ,

wherea,bare non-negative constants with a+b>_0, gand f satisfy g(x) = (₁+|x|²)⁻

2∗ s−q

2∗

s , for allx∈ _Rⁿ_, and

f(x) = (1+|x|²)^−1ξ, for all x∈_Rⁿ. Obviously, g ∈ L

2∗ s 2∗

s−q(_Rⁿ)∩L^∞_loc(_Rⁿ)and f ∈ L^ξ(_Rⁿ). Thus, they satisfy conditions(Z₁)and (Z₂). It is clearly that(M₃)hold, and for eachσ>0

M(t) =a+bt^θ−1≥a+bσ^θ−1=:m(σ)>0 for allt≥ σ, and

M(f t) =

Z _t

0

M(s)ds≥ ¹

θM(t)t for allt≥0.

Therefore,(M₁)and(M₂)hold. Then, Theorem1.3implies that the above problem admits one non-trivial non-negative solutions inW_K^s,p(_Rⁿ).

The framework of this paper is as follows. Section 2 introduces the necessary definitions and properties of spaceW_K^s,p(_Rⁿ). In Section 3, we give the proofs of the main results.

2 Variational framework

In this section, we first recall the basic variational frameworks and main lemmas for the the problem (1.1). Let L^q(_Rⁿ,g)be the weighted Lebesgue space, endowed with the norm

kvk^q_q,g=

Z

Rⁿg(x)|v(x)|^qdx.

The Banach space L^q(_Rⁿ,g) = (L^q(_Rⁿ,g),k · k_q,g)is uniformly convex according to Proposi- tion A.6 of [1]. Moreover, it follows from Lemma 2.1 of [3] that the embedding W_K^s,p(_Rⁿ) ,→ L^q(_Rⁿ_,g)is compact, so we have

kvk_q,g ≤C_gkvk_s,p for all v∈W_K^s,p(_Rⁿ), (2.1)

and C_g = S^−1pkgk

1 q

q1 > 0, where S = S(n,p,s) is the best fractional critical Sobolev constant, given by

S= inf

v∈W_K^s,p(_Rⁿ)\{0}

kvk^p_s,p kvk^p

L^p∗s(_Rⁿ)

. (2.2)

Obviously, from the above expression ofSand (1.2), we can get the fractional Sobolev embed- ding inequality that we know well:

kvk_Lp∗

s(_Rⁿ) ≤S^−1pkvk_s,p for all v∈W_K^s,p(_Rⁿ). (2.3) Similar to [24], we can also get that (W_K^s,p(_Rⁿ)_,kvk_s,p) is a uniformly convex Banach space.

Hence, it is a reflexive Banach space. Forv∈W_K^s,p(_Rⁿ), we define I_µ(v) = ¹

pMf _{Z Z}

R²ⁿ|v(x)−v(y)|^pK(x−y)dxdy

− ^µ q Z

Rⁿg(x)|v|^qdx

− ¹ p^∗_s

Z

Rⁿ|v|^p∗sdx−µ Z

Rⁿ f(x)vdx.

Clearly, by assumptions (1.2) and (Z₁)–(Z2), the energy functional I_µ : W_K^s,p(_Rⁿ) → _R as- sociated with the problem (1.1) is well defined. Obviously, the functional I_µ is of class C¹(W_K^s,p(_Rⁿ))and

hI_µ⁰(_v)_,ui=M(kvk_s,p^p )

Z Z

R²ⁿ|v(_x)−v(_y)|^p−2(_v(_x)−v(_y))(_u(_x)−u(_y)_K(_x−y)_dxdy

−µ Z

Rⁿg(x)|v|^q−2vudx−

Z

Rⁿ|v|^p∗s−2vudx−µ Z

Rⁿ f(x)udx,

for allv,u∈W_K^s,p(_Rⁿ), see for example ([3]:lemmas 4.2 ) with slight changes. Thus, the critical points of functionalI_µare weak solutions of the problem (1.1).

3 Proof of main results

In this section, we prove the main results of this article. For convenience, we use k · k_q to represent the norm of Lebesgue space L^q(_Rⁿ).

Lemma 3.1. If(M₁)–(M₂)and(Z₁)–(Z₂)hold. Then, there existρ,α= α(ρ),τ₀,µ₁ >0, such that I_µ(v)≥ αfor anyµ∈(_0,µ₀)withkvk_s,p=ρandkfk ≤τ₀, for allµ∈(_0,µ₁).

Proof. From(M₁), we get

M(f t)≥M(f 1)t^θ for allt ∈[0, 1]. (3.1) According to the Hölder inequality and (2.3), we have

Z

Rⁿg(x)|v|^qdx

≤ kgk_q1kvk^q_p∗

s ≤S^−qpkgk_q1kvk^q_s,p for allv∈W_K^s,p(_Rⁿ), (3.2) whereq₁ = _p∗^p∗s

s−q. In the same way, Z

Rⁿ|v|^p∗sdx

=kvk^p_p^∗s∗

s ≤S^−p

∗s

p kvk_s,p^p∗s (3.3)

and

Z

Rⁿ f(x)vdx

≤ kfk_ξkvk_p^∗_s ≤S^−1pkfk_ξkvk_s,p ≤εkvk^pθ_s,p+C_εkfk^(pθ)

0

ξ (3.4)

for all v ∈ W_K^s,p(_Rⁿ), where ξ = _p∗^p∗s

s−1. ε > 0, Cε > 0. Hence, from (3.1)–(3.4), we get for all v∈W_K^s,p(_Rⁿ)withkvk_s,p ≤1

I_µ(v)≥ M(^f 1)

p kvk^pθ_s,p−µS^−qpkgk_q1kvk^q_s,p−S^−p

∗s

p kvk_s,p^p∗s −µεkvk_s,p^pθ −µCεkfk^(pθ)

0 ξ

≥ M(^f 1)

2p kvk^pθ_s,p−µS^−qpkgk_q1kvk^q_s,p−S^−p

∗s

p kvk_s,p^p∗s −µCεkfk^(pθ)

0

ξ , (3.5)

with 0<εµ< ^M(f_2p¹⁾,(pθ)⁰= _pθ^pθ₋₁. Set

h(z) =µS^−qpkgk_q1z^q−pθ+S⁻

p∗

ps z^p∗s−pθ for allz>0. (3.6) It suffices to prove that h(z₀) < ^M(f_2p¹⁾ _{for some} z₀ = kvk_s,p > 0. Obverse that h(z) → _{∞ as} z→0⁺. Then,hhas a minimum atz₀ >0. To obtainz₀, we get

h⁰(z) =µS^−qp(q−pθ)kgk_q1z^q−pθ−1+S^−p

∗s

p (p^∗_s −pθ)z^{p∗s−pθ−1}. Leth⁰(z0) =0, we have

z₀=



 µS⁻

q

p(pθ−q)kgk_q1 S^−p

∗s

p (p^∗_s −pθ)





1 p∗

s−q

=µ

1 p∗

s−qS

1 p∗

s−q

0 ∈(0, 1],

whereS₀=S^−qp(pθ−q)kgk_q1/S^−p

∗s

p (p^∗_s −pθ)andµ≤S⁻₀¹. Furthermore,h(z₀)< ^M(f_2p¹⁾ means that

h(z₀) =S^−qp(pθ−q)kgk_q1(p^∗_s−q)(p^∗_s−pθ)⁻¹µ

p∗ s−pθ p∗

s−q S

q−pθ p∗

s−q

0 < M(^f 1)

2p . (3.7)

Then, by (3.5) and (3.7), there exist µ₁,τ0,α > 0 such that I_µ(v) ≥ α, with µ ∈ (0,µ₁), ρ=z₀=kvk_s,p andkfk_ξ ≤t₀ for every f ∈ L^ξ(_Rⁿ). Thus, the Lemma3.1is complete.

Next, we verify the compactness condition. For the convenience of the reader, we give the following definition.

Definition 3.2. Let I_µ ∈ C¹(X,R), we say that I_µ satisfies the (PS)_c condition at the level c∈ _R, if any sequence{v_j} ⊂X such that

I_µ(v_j)→c, I_µ⁰(v_j)→0 as j→_∞, (3.8) possesses a convergent subsequence inX.

Lemma 3.3. If(M₁)–(M₂)and(Z₁)–(Z₂)hold. Set c< 0. Then, there existsµ₂ such that for any µ∈ (_0,µ₂), the functionalI_µ fulfills(PS)_ccondition.

Proof. Consideringµ₂>0 small enough such that 1

pθ − ¹ p^∗_s

−_p∗^q

s−q

1 q− ¹

pθ

µ₂kgk_q1 ^p

∗s p∗

s−q

−

1− ¹ pθ

µ₂kfk_ξkvk_p^∗_s

<_min



 1

pθ − ¹

p^∗_s a₀^1θS ^p

∗sθ p∗

s−pθ

, 1

pθ − ¹ p^∗_s

(mS)

p∗ s p∗

s−p



, (3.9)

where q< p< pθ < p^∗_s, a₀ comes from(M₃),m= m(1)is defined in(M₂)withσ =1, while S is given in (2.2). Set µ ∈ (0,µ₂)and suppose that {v_j}_j be a (PS)_c sequence in W_K^s,p(_Rⁿ). Due to the degenerate nature of the equation (1.1), we will discuss it in two cases.

Case 1. inf_j∈Nkv_jk_s,p = d_µ > 0. First, we prove that the sequence {v_j}_j is bounded in W_K^s,p(_Rⁿ). From(M2), withσ=d_µ^p, there existsm=m(d_µ^p)>0 such that

M(kv_jk_s,p^p )≥m for all j∈_N. (3.10) Moreover, it follows from (M₁), (3.10) and (2.1) that

I_µ(v_j)− ¹

p^∗_shI_µ⁰(v_j)_,v_ji= ¹

pM(kf v_jk_s,p^p )− ¹

p^∗_sM(kv_jk_s,p^p )kv_jk_s,p^p − 1

q− ¹ p^∗_s

µkv_jk^q_q,g

−

1− ¹ p^∗_s

µ

Z

Rⁿ f(x)v_jdx

≥ 1

pθ − ¹ p^∗_s

mkv_jk_s,p^p − 1

q− ¹ p^∗_s

µC^q_gkv_jk^q_s,p

−

1− ¹ p^∗_s

µS^−1pkfk_ξkv_jk_s,p. (3.11) Therefore, from (3.8) there existsζ >0 andη>0 such that asj→_∞

c+ζkv_jk^q_s,p+ηkv_jk_s,p+o(1)≥ 1

pθ − ¹ p^∗_s

mkv_jk_s,p^p , (3.12) with q< p< pθ < p^∗_s. It means that{v_j}_j is bounded inW_K^s,p(_Rⁿ).

Taking into account the above fact and Lemmas 2.1, 2.4 of [3], there existv∈W_K^s,p(_Rⁿ)and βµ ≥0 such that, up to a subsequence still relabeled{v_j}_j, it follows that

v_j *v inW_K^s,p(_Rⁿ)_, kv_jk_s,p→ β_µ, v_j *v in L^p∗s(_Rⁿ), kv_j−vk_p^∗_s →ω, v_j →v in L^q(_Rⁿ,g), v_j →v a.e.in Rⁿ.

(3.13)

Further, by the above formula and Proposition A.8 of [1], we have

|v_j|^p∗s−2v_j *|v|^p∗s−2v in L^(p∗s)0(_Rⁿ), |v_j|^q−2v_j *|v|^q−2v in L^q0(_Rⁿ,g). (3.14) Discussion similar to Lemma 3.1 in reference [27], we can easily obtain thatvis a critical point of theC¹(W_K^s,p(_Rⁿ))functional

Ie_µ(v) = ¹

pM(β_µ^p)kvk_s,p^p −^µ

qkvk^q_q,g− ¹ p^∗_skvk^p_p^∗s∗

s −µ

Z

Rⁿ f(x)vdx. (3.15)

By (3.13), we get

jlim→_∞ Z

Rⁿg(x)(|v_j(x)|^q−2v_j(x)− |v(x)|^q−2v(x))(v_j(x)−v(x))dx=0. (3.16) Moreover, by again (3.13) and the well-known Brézis and Lieb lemma of [2], we have

kv_jk_s,p^p =kv_j−vk^p_s,p+kvk^p_s,p+o(1), kv_jk^p_p^∗s∗

s =kv_j−vk^p_p^∗s∗

s +kvk^p_p^∗s∗

s +o(1) (3.17) asj→∞. In particular, (3.8), (3.13), (3.15), (3.16) and (3.17) imply that as j→ _∞

o(1) =hI_µ⁰(v_j)−Ie_µ⁰(v),v_j−vi

=M(kv_jk_s,p^p )kv_jk^p_s,p+M(β_µ^p)kvk^p_s,p− hv_j,vi_s,p[M(kv_jk_s,p^p ) +M(β^p_µ)]

−µ Z

Rⁿg(x)(|v_j(x)|^q−2v_j(x)− |v(x)|^q−2v(x))(v_j(x)−v(x))dx

−

Z

Rⁿ(|v_j(_x)|^p∗s−2v_j(_x)− |v(_x)|^p∗s−2v(_x))(_vj(_x)−v(_x))_dx

=M(β^p_µ)(β^p_µ− kvk_s,p^p )− kv_jk^p_p^∗s∗

s +kvk^p_p^∗s∗

s +o(1)

=M(β^pµ)kv_j−vk^p_s,p− kv_j−vk^p_p^∗s∗

s +o(1), where hv_j,vi_s,p = R R

R²ⁿ|v_j(x)−v_j(y)|^p−2(v_j(x)−v_j(y))(v(x)−v(y)K(x−y)dxdy. Thus, we obtain the crucial formula

M(β_µ^p)lim

j→_∞kv_j−vk_s,p^p = lim

j→_∞kv_j−vk^p_p^∗s∗

s. (3.18)

Combining (2.2), (3.13) and (3.18), we have ω^p

∗s ≥SM(β_µ^p)ω^p. (3.19)

Ifω = 0, thanks to βµ > 0 and M admits a unique zero at 0, then (3.18) yields at once that v_j →vinW_K^s,p(_Rⁿ), concluding the proof. Instead, suppose thatω >0. Observing that (3.17), we can get

M(β^p_µ)(β^p_µ− kvk_s,p^p ) =ω^p

∗s, By (3.19), we obtain that

(ω^p

∗s)^psn = M(β^p_µ)^psn(β^p_µ− kvk_s,p^p )^psn ≥SM(β_µ^p)_{. (3.20)} Because we do not know the exact behavior of M, we have to think about both of these scenarios: either 0< β_µ<1 orβ_µ≥1. To do this, we separate the certificate the two subcases in the first case.

Subcase 1. 0< β_µ <1. It follows from (3.20) and(_M3)_that β

p2s n

µ ≥(β^p_µ− kvk_s,p^p )^psn ≥SM(β_µ^p)^n−nps ≥a

n−ps n

0 Sβ

p(θ−1)(n−ps) n

µ

and consideringn< ps_θ−^θ₁ = psθ⁰, it can be seen that β_µ^p ≥a

n−ps n

0 S_psθ−nⁿ₍

θ−1)

. (3.21)

Indeed, this limitation ⁿ

pθ⁰ < s can be derived directly from this fact 1 < θ < ^p_p^∗s = _n−ⁿ_ps. Making use of(_M3), (3.20) and (3.21), we get

ω^p

∗s ≥SM(β^p_µ)

n

ps ≥Sa₀β_µ^p(θ−1) _psⁿ

≥a₀^1θS_psθ−^nθ_n(

θ−1)

. (3.22)

Now, using (M₁)for any j∈_Nwe get I_µ(v_j)− ¹

pθhI_µ⁰(v_j)_,v_ji

= ¹

pM(kf v_jk_s,p^p )− ¹

pθM(kv_jk_s,p^p )kv_jk_s,p^p − 1

q− ¹ pθ

µkv_jk^q_q,g− 1

p^∗_s − ¹ pθ

kv_jk^p_p^∗s∗

s

−

1− ¹ pθ

µ

Z

Rⁿ f(x)v_jdx

≥ 1

pθ −¹ q

µkv_jk^q_q,g+ 1

pθ − ¹ p^∗_s

kv_jk^p_p^∗s∗

s −

1− ¹

pθ

µ Z

Rⁿ f(x)v_jdx.

For this, as j→ ∞, it follows from (3.8), (3.13), (3.17),(Z₁), the Hölder inequality and Young inequality that

c≥ 1

pθ − ¹ p^∗_s

ω^p

∗s +kvk^p_p^∗s∗ s

− 1

q− ¹ pθ

µkvk^q_q,g−

1− ¹ pθ

µ

Z

Rⁿ f(x)vdx

≥ 1

pθ − ¹ p^∗_s

ω^p

∗s +kvk^p_p^∗s∗ s

− 1

q− ¹ pθ

µkgk_q1kvk^q_p∗

s −

1− ¹

pθ

µkfk_ξkvk_p^∗_s

≥ 1

pθ − ¹ p^∗_s

ω^p

∗s +kvk^p_p^∗s∗ s

− 1

pθ − ¹ p^∗_s

kvk^p_p^∗s∗

s

− 1

pθ − ¹ p^∗_s

−_p∗^q

s−q

1 q− ¹

pθ

µkgk_q1 ^p

∗s p∗

s−q

−

1− ¹ pθ

µkfk_ξkvk_p^∗_s. (3.23) Finally, according to (3.22) we have

0>_c≥ 1

pθ − ¹ p^∗_s a

1 θ

0S ^p

∗sθ p∗

s−pθ

− 1

pθ − ¹ p^∗_s

−_p∗^q

s−q

1 q− ¹

pθ

µkgk_q1 ^p

∗s p∗

s−q

−

1− ¹ pθ

µkfk_ξkvk_p^∗_s >0,

where the above inequality by (3.9). In this subcase, we get our contradiction concluding the proof.

Subcase 2. β_µ ≥1. Here, it follows from (3.20) and(M₂)_withσ=_{1, we get} ω^p

∗s ≥(mS)^psn , with m=m(1)>0. Hence, by (3.23), we get

0>c≥ 1

pθ − ¹ p^∗_s

(mS)

p∗ s p∗

s−p − 1

pθ − ¹ p^∗_s

−_p∗^q

s−q

1 q− ¹

pθ

µkgk_q1 ^p

∗s p∗

s−q

−

1− ¹ pθ

µkfk_ξkvk_p^∗_s >0,

where again the above inequality by (3.9). We get a contradiction which completes for the proof of the fist case.

Case 2. inf_j∈_Nkv_jk_s,p =0. Thinking about two cases at zero. When 0 is an accumulation point of the real sequence {kv_jk_s,p}_j, and so there is a subsequence of {v_j}_j strongly converging to v = 0. This case can not happen thanks to it means that the trivial solution is a critical point at level c. When 0 is an isolated point of {v_j}_j, this case can happen, so that there is a subsequence still denoted by{kv_jkk_s,p}_k, such that inf_k∈_Nkv_jkk_s,p = d_µ >0 and we can prove as before. The proof of the second case is complete.

Proof of Theorem1.2. Throughout this paper, consideringµ₀ =_min{µ₁,µ₂}. Now, we look for the solutionv_∞.

Case 1. f 6≡ 0. We first show that there exists a φ₂ ∈ C^∞₀ (_Rⁿ), with kφ₂k_s,p = 1 such that R

Rⁿ f(x)φ₂(x)dx >0. Indeed, sinceC^∞₀ (_Rⁿ)is dense in L^ξ0(_Rⁿ)and|f|^ξ−2f ∈ L^ξ0(_Rⁿ). Then, there existsj0 >0 such that

kf_j0− |f|^ξ−2fk

L^ξ0(_Rⁿ)≤ ¹

2kfk^ξ−1

L^ξ(_Rⁿ). Hence, we obtain

Z

Rⁿ f(x)f_j0(x)dx≥ −kf_j0− |f|^ξ−2fk

L^ξ0(_Rⁿ)kfk_Lξ(_Rⁿ)+

Z

Rⁿ|f(x)|^ξdx≥ ¹ 2

Z

Rⁿ|f(x)|^ξdx>0.

Obviously, f_j0 ∈W_K^s,p(_Rⁿ). Letφ₂ = _k ^fj0

fj0ks,p, we have thatR

Rⁿ f(x)φ₂(x)dx>0. Like that I_µ(tφ₂) = ¹

pM(kf tφ₁k_s,p^p )− ^µt

q

q Z

Rⁿg(x)|φ₂|^qdx− ^t

p^∗_s

p^∗_s Z

Rⁿ|φ₂|^p∗sdx−µt Z

Rⁿ f(x)φ₂dx

≤

0max≤$≤1M($)

p t^p−^µt

q

q Z

Rⁿg(x)|φ₂|^qdx− ^t

p^∗_s

p^∗_s Z

Rⁿ|φ₂|^p∗sdx−µt Z

Rⁿ f(x)φ₂dx

<0 (3.24)

for smallt ∈(0, 1), on account ofp,q,p^∗_s >1.

Case 2. f ≡ 0. It is clearly that (3.24) still holds with kφ₂k_s,p = 1 for small enought ∈ (0, 1), because of 1< q< p< p^∗_s. Therefore, for any open ballB_R⊂W_K^s,p(_Rⁿ), we know

−_∞< cr = inf

v∈Be_R

I_µ(v)<0. (3.25)

Hence,

c= inf

v∈Beρ

I_µ(v)<0 and inf

v∈∂B_ρI_µ(v)>0, (3.26) whereρ>0 is given lemma3.1. Setε_j ↓0 such that

0<ε_j < inf

v∈∂Bρ

I_µ(v)− inf

v∈Bρ

I_µ(v). (3.27)

So, by Ekeland’s variational principle in [8], there exists {v_j}_j ⊂ Be_ρsuch that

c≤ I_µ(v_j)≤c+ε_j (3.28)